Preheat your oven to 350°F (175°C). Grease and flour a baking pan.
In a mixing bowl, cream together the butter and sugar until light and fluffy.
Add the eggs one at a time, beating well after each addition.
Stir in the applesauce.
In another bowl, whisk together the flour, baking soda, cinnamon, nutmeg, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Pour the batter into the prepared baking pan.
Bake for 45 minutes, or until a toothpick inserted into the center comes out clean.
Allow the cake to cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.